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Stochastic Complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 and is a generalization of classical information theory. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program ''P'' computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than ''P'''s own leng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal (programming Language)
Pascal is an imperative and procedural programming language, designed by Niklaus Wirth as a small, efficient language intended to encourage good programming practices using structured programming and data structuring. It is named after French mathematician, philosopher and physicist Blaise Pascal. Pascal was developed on the pattern of the ALGOL 60 language. Wirth was involved in the process to improve the language as part of the ALGOL X efforts and proposed a version named ALGOL W. This was not accepted, and the ALGOL X process bogged down. In 1968, Wirth decided to abandon the ALGOL X process and further improve ALGOL W, releasing this as Pascal in 1970. On top of ALGOL's scalars and arrays, Pascal enables defining complex datatypes and building dynamic and recursive data structures such as lists, trees and graphs. Pascal has strong typing on all objects, which means that one type of data cannot be converted to or interpreted as another without explicit conversions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Discovery
The concept of multiple discovery (also known as simultaneous invention) is the hypothesis that most scientific discoveries and inventions are made independently and more or less simultaneously by multiple scientists and inventors. The concept of multiple discovery opposes a traditional view—the "heroic theory" of invention and discovery. Multiple discovery is analogous to convergent evolution in biological evolution. Multiples When Nobel laureates are announced annually—especially in physics, chemistry, physiology, medicine, and economics—increasingly, in the given field, rather than just a single laureate, there are two, or the maximally permissible three, who often have independently made the same discovery. Historians and sociologists have remarked the occurrence, in science, of "multiple independent discovery". Robert K. Merton defined such "multiples" as instances in which similar discoveries are made by scientists working independently of each other. Merton con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmic Probability
In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s. It is used in inductive inference theory and analyses of algorithms. In his general theory of inductive inference, Solomonoff uses the method together with Bayes' rule to obtain probabilities of prediction for an algorithm's future outputs. In the mathematical formalism used, the observations have the form of finite binary strings viewed as outputs of Turing machines, and the universal prior is a probability distribution over the set of finite binary strings calculated from a probability distribution over programs (that is, inputs to a universal Turing machine). The prior is universal in the Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated. Formally, the probability P is not a probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ray Solomonoff
Ray Solomonoff (July 25, 1926 – December 7, 2009) was an American mathematician who invented algorithmic probability, his General Theory of Inductive Inference (also known as Universal Inductive Inference),Samuel Rathmanner and Marcus Hutter. A philosophical treatise of universal induction. Entropy, 13(6):1076–1136, 2011. and was a founder of algorithmic information theory. He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability. He circulated the first report on non-semantic machine learning in 1956."An Inductive Inference Machine", Dartmouth College, N.H., version of Aug. 14, 1956(pdf scanned copy of the original)/ref> Solomonoff first described algorithmic probability in 1960, publishing the theorem that launched Kolmogorov complexity and algorithmic information theory. He first described these results at a conference at Caltech in 1960, and in a report, Feb. 1960, "A Preliminary Report on a General Theory of I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Structure
In computer science, a data structure is a data organization and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the relationships among them, and the Function (computer programming), functions or Operator (computer programming), operations that can be applied to the data, i.e., it is an algebraic structure about data. Usage Data structures serve as the basis for abstract data types (ADT). The ADT defines the logical form of the data type. The data structure implements the physical form of the data type. Different types of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. For example, Relational database, relational databases commonly use B-tree indexes for data retrieval, while compiler Implementation, implementations usually use hash tables to look up Identifier (computer languages), identifiers. Data s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpreter (computing)
In computer science, an interpreter is a computer program that directly executes instructions written in a programming or scripting language, without requiring them previously to have been compiled into a machine language program. An interpreter generally uses one of the following strategies for program execution: # Parse the source code and perform its behavior directly; # Translate source code into some efficient intermediate representation or object code and immediately execute that; # Explicitly execute stored precompiled bytecode made by a compiler and matched with the interpreter's virtual machine. Early versions of Lisp programming language and minicomputer and microcomputer BASIC dialects would be examples of the first type. Perl, Raku, Python, MATLAB, and Ruby are examples of the second, while UCSD Pascal is an example of the third type. Source programs are compiled ahead of time and stored as machine independent code, which is then linked at run-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, " is unique up to " means that all objects under consideration are in the same equivalence class with respect to the relation . Moreover, the equivalence relation is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation that relates two lists if one can be obtained by reordering (permutation, permuting) the other. As another example, the statement "the solution to an indefinite integral is , up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix Code
A prefix code is a type of code system distinguished by its possession of the prefix property, which requires that there is no whole Code word (communication), code word in the system that is a prefix (computer science), prefix (initial segment) of any other code word in the system. It is trivially true for fixed-length codes, so only a point of consideration for variable-length code, variable-length codes. For example, a code with code has the prefix property; a code consisting of does not, because "5" is a prefix of "59" and also of "55". A prefix code is a uniquely decodable code: given a complete and accurate sequence, a receiver can identify each word without requiring a special marker between words. However, there are uniquely decodable codes that are not prefix codes; for instance, the reverse of a prefix code is still uniquely decodable (it is a suffix code), but it is not necessarily a prefix code. Prefix codes are also known as prefix-free codes, prefix condition codes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariance Theorem
{{disambiguation, math ...
Invariance theorem may refer to: *Invariance of domain, a theorem in topology *A theorem pertaining to Kolmogorov complexity *A result in classical mechanics for adiabatic invariants *A theorem of algorithmic probability See also *Invariant (mathematics) In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-code
In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actions and conditions. Although pseudocode shares features with regular programming languages, it is intended for human reading rather than machine control. Pseudocode typically omits details that are essential for machine implementation of the algorithm, meaning that pseudocode can only be verified by hand. The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation. The reasons for using pseudocode are that it is easier for people to understand than conventional programming language code and that it is an efficient and environment-independent description of the key principles of an algorithm. It is commonly used in textbooks and scientific publications to document a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
